Large Salem Sets Avoiding Nonlinear Configurations

TL;DR

This paper constructs large Salem sets in the torus that avoid certain nonlinear configurations, extending previous work on pattern avoidance with large Hausdorff dimension.

Contribution

It introduces new methods to build Salem sets avoiding nonlinear solutions for a broad class of functions, with explicit dimension bounds.

Findings

Constructed Salem sets avoiding solutions to nonlinear equations with specified dimensions.

Extended pattern avoidance to uncountable families of Lipschitz functions.

Achieved Salem sets with dimensions depending on the number of variables and function properties.

Abstract

We construct large Salem sets avoiding patterns, complementing previous constructions of pattern avoiding sets with large Hausdorff dimension. For a (possibly uncountable) family of uniformly Lipschitz functions $\{ f_i : (\mathbb{T}^d)^{n-2} \to \mathbb{T}^d \}$, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-1)$ avoiding nontrivial solutions to the equation $x_n - x_{n-1} = f_i(x_1,\dots,x_{n-2})$. For a countable family of smooth functions $\{ f_i : (\mathbb{T}^d)^{n-1} \to \mathbb{T}^d \}$ satisfying a modest geometric condition, we obtain a Salem subset of $\mathbb{T}^d$ with dimension $d/(n-3/4)$ avoiding nontrivial solutions to the equation $x_n = f(x_1,\dots,x_{n-1})$. For a set $Z \subset \mathbb{T}^{dn}$ which is the countable union of a family of sets, each with lower Minkowski dimension $s$, we obtain a Salem subset of $\mathbb{T}^d$ of dimension $(dn - s)/(n - 1/2)$ whose Cartesian product does not intersect $Z$ except at points with non-distinct coordinates.